Indestructibility of Vopěnka’s Principle

Vopenka's Principle is a natural large cardinal axiom that has recently found applications in category theory and algebraic topology. We show that
Vopenka's Principle and Vopenka cardinals are relatively consistent with a broad
range of other principles known to be independent of standard (ZFC) set theory,
such as the Generalised Continuum Hypothesis, and the existence of a denable
well-order on the universe of all sets. We achieve this by showing that they are indestructible under a broad class of forcing constructions, specically, reverse Easton
iterations of increasingly directed closed partial orders.